© yamacraw, fall 2002 power efficient range assignment in ad-hoc wireless networks e. althous (mpi)...
DESCRIPTION
© Yamacraw, Fall 2002 Asymmetric Connectivity Strongly connected Nodes transmit messages within a range depending on their battery power, e.g., a b c b,d g f,e,d,a a b d g f e c b a c d g f e Range radii Message from “a” to “b” has multi-hop acknowledgement route a b d g f e c 1 1 1TRANSCRIPT
© Yamacraw, Fall 2002
Power Efficient Range Assignment Power Efficient Range Assignment in Ad-hoc Wireless Networksin Ad-hoc Wireless Networks
E. Althous (MPI)E. Althous (MPI)G. Calinescu (IL-IT)G. Calinescu (IL-IT)I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)
S. Prasad (GSU)S. Prasad (GSU)N. Tchervinsky (IL-IT)N. Tchervinsky (IL-IT)A. Zelikovsky (GSU)A. Zelikovsky (GSU)
ES0036ES0036
© Yamacraw, Fall 2002
Ad Hoc Wireless NetworksAd Hoc Wireless Networks• Applications in battlefield, disaster relief, etc.• No wired infrastructure• Battery operated power conservation critical• Omni-directional antennas + Uniform power detection
thresholdsTransmission range = disk centered at the node
• Signal power falls inversely proportional to dk
Transmission range radius = kth root of node power
© Yamacraw, Fall 2002
Asymmetric ConnectivityAsymmetric Connectivity
Strongly connected
Nodes transmit messages within a range depending on their battery power, e.g., ab cb,d gf,e,d,a
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Range radii
Message from “a” to “b” has multi-hop acknowledgement route
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© Yamacraw, Fall 2002
Symmetric ConnectivitySymmetric Connectivity
• Per link acknowledgements symmetric connectivity• Two nodes are symmetrically connected iff they are within transmission
range of each other
Node “a” cannot get acknowledgement directly from “b”
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Asymmetric Connectivity
Increase range of “b” by 1 and decrease “g” by 2
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© Yamacraw, Fall 2002
Min-power Symmetric Connectivity ProblemMin-power Symmetric Connectivity Problem• Given: set S of nodes (points in Euclidean plane), and coefficient k• Find: power levels for each node s.t.
– There exist symmetrically connected paths between any two nodes of S
– Total power is minimized
Power assigned to a node = largest power requirement of incident edges
k=2 total power p(T)=257a
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Power levels for k=2
Distances
© Yamacraw, Fall 2002
ResultsResults
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• Previous results– Max power objective
• MST is optimal [Lloyd et al. 02]– Total power objective
• NP-hardness [Clementi,Penna&Silvestri 00] • MST gives factor 2 approximation [Kirousis et al. 00]
• Our results – General graph formulation– Improved approximation results
• 5/3 + • 11/6 for a practical greedy algorithm
– New ILP formulation– Several swapping heuristics– Experimental study
© Yamacraw, Fall 2002
Graph FormulationGraph Formulation
Power cost of a node = maximum cost of the incident edge Power cost of a tree = sum of power costs of its nodes
Min-Power Symmetric Connectivity Problem in Graphs: Given: edge-weighted graph G=(V,E,c), where c(e) is the power required to
establish link eFind: spanning tree with a minimum power costd
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Power costs of nodes arePower costs of nodes are yellow yellowTotal power cost of the tree isTotal power cost of the tree is 68 68
© Yamacraw, Fall 2002
MST AlgorithmMST AlgorithmTheorem: The power cost of the MST is at most 2 OPTProof
(1) power cost of any tree is at most twice its cost p(T) = u maxv~uc(uv) u v~u c(uv) = 2 c(T)
(2) power cost of any tree is at least its cost
(1) (2)
p(MST) 2 c(MST) 2 c(OPT) 2 p(OPT)
1+ 1+ 1+
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Power cost of MST is n Power cost of OPT is n/2 (1+ ) + n/2 n/2
n points
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Greedy Fork Contraction AlgorithmGreedy Fork Contraction AlgorithmFork F is the set of two adjacent edgesGain of fork F, gain(F), is by how much inserting of F and removing other
two edges improves the power cost
Input: Graph G=(V,E,cost) with edge costs
Output: Low power-cost tree spanning VTMST(G)H
Repeat foreverFind fork F with maximum gainIf gain(F) is non-positive, exit loopHH U FTT/F
Output T H
© Yamacraw, Fall 2002
Edge Swapping HeuristicEdge Swapping Heuristic
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Remove edge 10 Remove edge 10 power cost decrease = -6power cost decrease = -6
Reconnect components with min increase in power-cost = +5Reconnect components with min increase in power-cost = +5
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For each edge do• Delete an edge• Connect with min increase in power-cost• Undo previous steps if no gain
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© Yamacraw, Fall 2002
Integer Linear Program FormulationInteger Linear Program Formulation
yuv = range variable, =1 if for uv is maximum weight edge from u in tree T
xuv = tree variable, =1 if uv is in tree T
- choose a single power range
- power range connects endpoints
- connectivity requirement
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Experimental StudyExperimental Study
• Random instances up to 100 points• Compared algorithms
– branch and cut based on novel ILP formulation [Althaus et al. 02]
– Greedy fork-contraction– Incremental power-cost Kruskal– Edge swapping– Delaunay graph versions of the above
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Percent Improvement Over MSTPercent Improvement Over MST
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Runtime (CPU seconds)Runtime (CPU seconds)
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Percent Improvement Over MSTPercent Improvement Over MST